Electron Density and Its Derivatives

Electron Density \(\rho\left(\mathbf{r}\right)\) represents the probability of observing an electron (possibly with specified spin) within a certain volume element \(d\boldsymbol{r}\) at \(\mathbf{r}\). It is most fundamentally defined through the wave-function:

\[\rho(\boldsymbol{r}) = N\int \ldots \int \vert \Psi(\boldsymbol{x}_1 , \boldsymbol{x}_2 , \ldots , \boldsymbol{x}_N) \vert^2 d\sigma_1 , d\boldsymbol{x}_2 , \ldots , d\boldsymbol{x}_N ,\]

where the vectors \(\boldsymbol{x}_i\) include the space coordinates, \(\boldsymbol{r}_i\), and the spin coordinate, \(\sigma_i\), of the i-th electron.

Shape Function \(\sigma\left(\mathbf{r}\right)\) represents the electron density per particle:

\[\sigma\left(\mathbf{r}\right) = \frac{\rho\left(\mathbf{r}\right)}{N} = \frac{\rho\left(\mathbf{r}\right)}{\int \rho\left(\mathbf{r}\right) d\mathbf{r}}\]

Electron Density Gradient \(\nabla\rho\left(\mathbf{r}\right)\) represents the 1st-order partial derivatives of the electron density with respect to the coordinates which is a vector:

\[\begin{split}\nabla \rho\left(\mathbf{r}\right) = \begin{bmatrix} \frac{\partial \rho\left(\mathbf{r}\right)}{\partial x} \\ \frac{\partial \rho\left(\mathbf{r}\right)}{\partial y} \\ \frac{\partial \rho\left(\mathbf{r}\right)}{\partial z}\end{bmatrix}\end{split}\]

The gradient points in the direction of steepest increase of the electron density at a point, and is zero at a position of a critical point of the electron density. The \(\nabla \rho\left(\mathbf{r}\right)\) is therefore a key ingredient for the topological analysis of the electron density.

Electron Density Gradient Norm \(\nabla\rho\left(\mathbf{r}\right)\) represents the norm of the electron density gradient vector:

\[\begin{split}\lvert \nabla\rho\left(\mathbf{r}\right) \rvert &= \sqrt{\nabla\rho\left(\mathbf{r}\right) \cdot \nabla\rho\left(\mathbf{r}\right)} \\ &= \sqrt{\left(\frac{\partial \rho\left(\mathbf{r}\right)}{\partial x}\right)^2 + \left(\frac{\partial \rho\left(\mathbf{r}\right)}{\partial y}\right)^2 + \left(\frac{\partial \rho\left(\mathbf{r}\right)}{\partial z}\right)^2}\end{split}\]

Electron Density Hessian \(\nabla\nabla^{T}\rho\left(\mathbf{r}\right)\) represents the 2nd-order partial derivatives of the electron density with respect to coordinates which is a symmetric matrix:

\[\begin{split}\nabla \nabla^{T} \rho\left(\mathbf{r}\right) = \begin{bmatrix} \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial x^2} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial x \partial y} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial x \partial z} \\ \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial y \partial x} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial y^2} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial y \partial z} \\ \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial z \partial x} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial z \partial y} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial z^2} \\ \end{bmatrix}\end{split}\]

The eigenvalues of the Hessian matrix are often used to classify critical points.

Electron Density Laplacian \(\nabla^2\rho\left(\mathbf{r}\right)\) represents the trace of the electron density Hessian matrix which is equal to sum of its \(\left(\lambda_1, \lambda_2, \lambda_3\right)\) eigenvalues:

\[\nabla^2 \rho\left(\mathbf{r}\right) = \nabla\cdot\nabla\rho\left(\mathbf{r}\right) = \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial x^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial y^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial z^2} = \lambda_1 + \lambda_2 + \lambda_3\]

It is perhaps the first quantity that was used to visualize shells in molecules. It has been used and interpreted as a electron-pair-region-locator at least since Richard Bader nailed a picture of the Laplacian of the electron density to Ron Gillespie’s office door!) The Laplacian of the electron density tends to be negative where electron density accumulates and positive elsewhere.

The gradient and Laplacian of the electron density have units of \(\text{(length)}^{–4}\) and \(\text{(length)}^{–5}\), respectively, and they increase systematically as the number of electrons increases. For this reason, one often defines the dimensionless gradient/Laplacian, which are often called the electron density reduced gradient/Laplacian. These are often useful for building density functionals in DFT and/or visualizing purposes that are suitable for both core and valence regions.

Reduced Density Gradient \(s\left(\mathbf{r}\right)\) represents the dimensionless form of the electron density gradient norm, where the pre-factor is chosen by convention, and is omitted in some other implementations.

\[s\left(\mathbf{r}\right) = \frac{1}{3\left(2\pi ^2 \right)^{1/3}} \frac{\lvert \nabla \rho\left(\mathbf{r}\right) \rvert}{\rho\left(\mathbf{r}\right)^{4/3}}\]

This is used, for example, in the non-covalent interactions analysis (NCI). The traditional pre-factor one uses is different depending on whether one uses spin-resolved or spin-unresolved treatments.

Reduced Density Hessian \(q\left(\mathbf{r}\right)\) represents the dimensionless form of the electron density Laplacian, where the pre-factor is chosen by convention, and is omitted in some other implementations.

\[q\left(\mathbf{r}\right) =\]

This is used, for example, in analyzing the shell structure and electron delocalization/metallicity. The traditional pre-factor one uses is different depending on whether one uses spin-resolved or spin-unresolved treatments.